Data Assimilation for 2-D Advection-Dispersion Equations

Transcription

1 Dt Asslton for -D Advecton-Dsperson qutons Sergey Kvv Insttute of Mthetcl Mchnes nd Syste Probles Ntonl Acdey of Scences of Ukrne 4 Glushkov pr. 387 Kev Ukrne slk@env.sp.kev.u Abstrct. Dt sslton bsed on vrtonl prncple for preter estton of -D dvecton-dsperson equtons s consdered. It s ssued tht pror esttons for the odel preters nd the ntl condton re vlble. Iproveent of preters for the rdonuclde trnsport odel s reduced to n optzton proble wth qudrtc cost functonl. he cost functonl coprses the esureent odel nd odel errors the ntl condton uncertnty nd preter penltes. he exstence of unque soluton for the stte equtons nd the djont syste s proved. Dfferentl propertes of the cost functonl re nvestgted nd necessry condton for cost functonl nu s derved. Restrctons on the cost functonl weghts whch gurntee the exstence of unque sttonry pont re derved. Introducton In ths pper we concentrte on preter estton for the dvecton-dsperson equtons whch for exple cn be used to descrbe rdonuclde trnsport by surfce wter flow. he rdonuclde trnsport n the queous phse nd on suspended sedents by surfce wter flow s governed by the equtons wth the snk-source ter descrbng physcl-checl nterctons nd eroson-deposton exchnge processes [-] ( hcl) cl s + ( vhc l) = he λhcl αshs( kdρ cl cs) t x x x b ( φ) z α k ρ ρ c c () b d b l b s ( hscs) Scs + ( vhscs) = he λhscs + αshs kdρ cl cs + t x x x ρb qc b b qc s s + () where h(xt) s the surfce wter depth; v (xt) re the surfce wter velocty coponents; φ(x) s the porosty of upper sol lyer; ρ s the densty of wter; ρ b (x) s the densty of sol trx; S(xt) s the suspended sedent concentrton; c l (xt) s the P.M.A. Sloot et l. (ds.): ICCS 3 LNCS 658 pp Sprnger-Verlg Berln Hedelberg 3

2 6 S. Kvv voluetrc rdonuclde ctvty n queous phse; c s (xt) s the rdonuclde ctvty on suspended sedent; c b (xt) s the voluetrc rdonuclde ctvty n upper sol lyer; z ( x t) s the thckness of ctve upper sol lyer; λ s the rdonuclde decy constnt; q s (xt) nd q b (xt) re the deposton nd eroson rtes; e (xt) the dffuson s b coeffcents; k d k d nd s b re the prtton coeffcents nd the exchnge rtes respectvely; =. Contnton of the ctve upper sol lyer s descrbed by the equtons b cb (( φ) zcb) = ( φ) ze + t x x b + φ αbz kdρbρ cl cb λz c b ρb qbcb + qsc s (3) s Vlues of the prtton coeffcents k d (x) k d b(x) nd exchnge rtes α s (x) α b (x) re deterned by the sol physcl-checl propertes nd fors of rdonuclde contnton. hese coeffcents chrcterze the nterchnge between the vrous rdonuclde fors. So forecst qulty of rdonuclde contnton of surfce reservors depends on ccurcy deternton of ther vlues. xperence of the Chernobyl ccdent hs shown tht exct vlues of the prtton coeffcent nd exchnge rtes re unknown just fter ccdent nd t the best ther esttons re only vlble. Nowdys dt sslton ethods used for preter estton s bsed on estton theory or control theory [3]. stton theory usully nvolves sequentl esttons of both odel nd dt errors by usng sttstcl pproch. Control theory ethods re generlly bsed on vrtonl prncples. In ths pper we consder vrtonl sslton pproch [4] for preter estton of the odel of rdonuclde trnsport by surfce wter flow. In ths cse preter estton proble reduces to nzton of qudrtc cost functonl on set of dssble vlues of the odel preters n neghborhood of ther pror esttons. On the set of dssble vlues of the odel preters the dfferentl propertes of the cost functonl re nvestgted necessry condton for nu of the cost functonl s derved nd suffcent condton for exstence of unque sttonry pont s obtned. Proble Stteent Rewrte the odel equtons ()-(3) n the vector for c ( Ac) + ( Ac ) = D G( µν ) c+ r t x x x ( xt ) Q ( ) =Ω (4) where ( ) ( ) c x t = c c c = ( c Sc c ) s the stte vector; r( x t) = ( r r r ) 3 l s b s b s the functon of the odel error; µ ( µ µ ) ( αs kd ρ αbkd ρb ρ) 3 = = nd ν =

3 Dt Asslton for -D Advecton-Dsperson qutons 6 ( ν ν ) ( α α ) = s b s the preter vector; Ω s connected bounded set n wth twce boundedly dfferentble boundry Ω; n s the n-densonl uclden spce. he functon of odel errors r(xt) we shll consder s n uxlry vrble whch s needed to nze. Mtrces A A D re dgonl trces Mtrx G(µν) s represented s where ( φ ) { 33 } { } A = dg = dg h h z ; { 33 } { } A = dg = dg v h v h ; { 33 } { b } D = dg d d d = dg he he ze. λ h+ % µ hs + % µ z % νh % νz G( µν ) = % µ hs λh+ % νh+ q q µ z q λz+ νz+ q % % z = z ; q = qs S ; q qb ρb = ; µ ν nd µ = µ ψ % = ν ψ x ; the functon ψ(x)=(ψ ψ ) stsfes to reltons ν % x ψ k ( x) x Ωk = x Ωk ; U k ; ΩI Ω j = j. k= Ω= Ω Let eleents of the trces A A D nd G re bounded esurble functons on Q. he trces A A nd D re dfferentble wth respect to t nd x nd the followng nequltes re vld kk d υ π kk π υ υυ > k = 3; = vr x A D A ( ) t = t = t xt Q vr x A D A ( xt ) Q = x = xk = x π π3 (5)

4 6 S. Kvv vr x A G π4 ( xt ) Q = u Ud n j j where A = ( ) s the uclden nor of n n-trx A { } =. j= For the equton (4) let s consder the frst boundry vlue proble wth hoogeneous boundry condton ( ) c x = c x x Ω (6) cxt ( ) = ( xt ) [ ] he esureent odel s descrbed by equton d Γ= Ω (7) c ( x) = H( x) c( x ) + ϑ( x) (8) where c d (x) s the observton dt; ϑ s the esureent error; H(x) s the observton trx eleents of whch re bounded nd esurble on Ω nd hve bounded dervtves wth respect to x. We shll ssue tht exct vlues of the odel preters nd the ntl condton re unknown nd ther pror esttons re only vlble = + µ µ µ ϑ ν = + c ( x) c ( x) ϑ ( x) ν ν ϑ = + c where µ nd ν re the gven pror esttons of the odel preters; c s the known pror estton of the ntl stte; ϑ µ nd ϑ ν re the errors of the odel preters; ϑ c s the error of the ntl stte. Denote by U the spce L( Q) W& ( Ω) wth eleents u=(u u u 3 u 4 ) =(µνrc ). Introduce n the spce U the followng sclr product uw = u w + u w + u w + u w U 3 3 L( Q) 4 4 W n k for uw U where ξ η n k= ξjηj s the sclr product n n k (further = j= we shll ot denson of the uclden spce n sclr product when t cn be done wthout bguty); L (Q) s the Hlbert spce of esurble functons on Q wth fnte sclr product f g f g dxdt L ( Q) =. Ω j ( Ω)

5 Dt Asslton for -D Advecton-Dsperson qutons 63 & ( Ω) s the subspce of the Hlbert spce W W Ω dense set n whch s set of ll nfntely dfferentble fnte functons on Ω. he sclr product n W defned by f g W f g x f g = Ω + x dx. = Ω Ω s Let U d s closed convex bounded set n U defnng dssble vlues of the odel preters (4) the ntl stte (6) nd the odel error. Further the set U d s clled s set of dssble odel preters. hen we shll consder the proble of proveent of the odel preters u=(µνrc ) for the equton (4) by usng the observtons (8) s proble of nzton the followng cost functonl on U d ( ( ; ) d ) { ( ; ) d I ucxtu = c x H x cxu} R { c ( x ) H( x) cxu ( ; ) } dx+ Ω ( ) ( ) + r Rrdxdt+ c x c x R c x c x dx+ Q Ω ( µ µ ) Rµ ( µ µ ) ( ν ν ) Rν ( ν ν ) + + = where R (x) R (xt) R (x) R µ R ν re postve-defnte syetrc weght trces. We shll ssue tht eleents of the trces R (x) R (x) R (xt) re bounded esurble functons on Ω nd Q; the functons c d (x)c (x) W& ( Ω). Moreover the trx R (x) hve bounded dervtves wth respect to x on Ω. We cn consder the weght trces s preters of dt sslton nd ther choce depends on users. Usng the ethod of the Lgrnge ultplers we get the followng djont syste (9) c c c A + A + D G ( µν ) c = t x x x d ( ) c x = A R H c Hc x Ω () () c ( x t) = ( ) xt Γ ()

6 64 S. Kvv 3 xstence of Soluton of the Stte nd Adjont Probles Consder syste of equtons w w w B() xt Bj() xt + B() xtw + C() xt + Cxtw () = f() xt t x x j x where w(xt) nd f(xt) re -densonl vector-functons; B B j B C nd C re -trces (B B j re the dgonl trces nd B j =B j ) wth eleents b kk(xt) kk kl b j (xt) b (xt) c kl(xt) nd c kl (xt) respectvely; x=(x x ). We shll ssue tht the followng reltonshps re vld σζ ζ kk bj ζζ j γζ ζ σ > k = j= (3) B σ > (4) σ ξ ξ ξ ξ γ ξ ξ =. he trces B B B j re dfferentble wth respect to t nd x nd B B j B vr x γ ( xt ) Q t j t t for ny rel ξ = ( ξ... ξ ) nd ζ ( ζ ζ ) B B vr x j B ( xt ) Q x j xk x γ (5) 3 vr x B 4 ( xt ) Q C C γ = =. Consder the frst ntl-boundry vlue proble for the equton (3) wth the followng ntl nd boundry condtons wx ( ) = φ ( x) x Ω; (6) wxt ( ) = ( xt ) Γ. (7) Defnton. he functon wxt ( ) W ( Q) s clled generlzed soluton of (3) (6)-(7) f for ll t fro [] nd ny vector-functon xt L ( Q) dentty s fulflled η( ) the followng nd t Ω ( j ) ( ) η Bwt Bjwx + Bw + Cwx + Cw+ f dxdt = x (8)

7 Dt Asslton for -D Advecton-Dsperson qutons 65 [ φ ] wx ( t) x dx s t +. (9) Ω W Q s the Hlbert spce of functons wth zero-vlue t Γ nd the sclr product s defned by wk vk wk vk wk v k w v = wkvk dxdt Q k= t t = x x j= x xj x x j ( ) W ( Q) heore. Let the coeffcents nd free ters n the equton (3) stsfy to restrctons (4)-(5). Ω s open connected bounded don n wth boundry Ω hvng second bounded dervtves. hen for n rbtrry functon φ (x) fro W& ( Ω) nd for n rbtrry functon f( x t) L Q W Q such tht nd for ll [ ] there s unque soluton of the proble (3) (6)-(7) n wxt M x f t ( ) W Q φ + L Q W ( Ω) wxt ( ) σ exp { γ5tσ } γ φ L ( x) f Ω + L ( Ω) L( Q) where the constnt M depends only on σ σ γ γ γ γ 3 γ 4 Ω nd γ5 = + γ + γ4 + γ4σ. he solvblty of the ntl-boundry vlue proble (4) (6)-(7) cn be proved by usng the Glerkn ethod n slr wy s n [5]. Corollry. Let the coeffcents of the equton (4) stsfy the restrctons (5) on U d. he trces H(x) R (x) re bounded nd esurble on Ω nd hve bounded dervtves wth respect to x nd c d ( x) W& Ω. hen for ech u ( µν r c ) dt unque soluton n = fro U d the probles (4) (6)-(7) nd ()-() W Q. 4 xstence of Cost Functonl Mnu heore. Let stte vector c(xt;u) s governed by the frst ntl-boundry vlue proble (4) (6)-(7) the coeffcents of whch stsfy to the restrctons (5). Let the set of dssble odel preters U d s closed convex bounded set n U. Cost functonl s defned by the reltonshp (9).

8 66 S. Kvv hen U = u = ( µν r c ) Ud : I ( u c( x t; u) ) = nf I( v c( x t; v) ) s nonepty nd wekly copct set. Any nzng sequence {u k } U d v Ud converges wekly to U *. Proof. Let {u k } U d s nzng sequence.e. k k ( u c( x t; u )) nf ( u c( x t; u) ) I I s k. () u Ud Accordng to the heore the proble (4) (6)-(7) dts unque soluton on U d such tht cxtu ( ; ) M W ( Q ) where the constnt M does not depend on u. Due to boundedness of the solutons c(xt;u) on U d nd boundedness of the set U d there s subsequence {u N } of the sequence {u k } tht c(xt;u N ) cxt %( ) wekly n W ( Q ) µ N µ% nd ν N ν% n c N c% wekly n W& ( Ω) r N r% wekly n L (Q). Moreover fro the beddng theores [6 pp.35-36] we hve tht c N converges to N c% n L (Ω) nd cxtu ( ; ) cxt %( ) s N unforly wth regrd to L ( Ω) t []. Snce U d s closed convex set the set U d s wekly closed nd ( µν % ) u% = % r% c% U d. It s esy to show tht cxt %( ) concdes wth cxtu% ( ; ). o show ths t s suffcently n (8)-(9) wrtten for the c(xt;u N ) to pss to the lt s N. Hence the functon cxt %( ) s soluton of the proble (4) (6)-(7) for u = u%. he functonl ( ucxtu ( ; )) I s convex contnuous functonl of the rguents so t s wekly lower secontnuous nd ( N N u c x t u ) ( u c x t ) l I ( ; ) I % %( ) N herefore fro () follows ( ucxt ( )) nf ( ucxtu ( ; )) I % % I. u U d % % nd u% U. u U d It rens for us to show tht the set U * s wekly copct. Let s tke n rbtrry sequence {u k } U *. hen {u k } U d nd there s subsequence {u N } whch converges N N wekly to soe pont u% Ud. But I ( u c( x t; u )) = nf I ( u c( x t; u) ) u Ud hus I ( ucxt ( )) = nf I( ucxtu ( ; ))

9 Dt Asslton for -D Advecton-Dsperson qutons 67 N= so {u N } s nzng sequence. As shown bove the {u N } converges wekly to U * therefore u% U. Hence the set U * s wekly copct tht copletes the proof. Let s rerk tht the set of solutons of the proble (4) (6)-(7) s non-convex therefore nu of the cost functonl cn be non-unque. heore 3. Let the ssuptons of heore re fulflled. hen the cost functonl (9) s contnuously dfferentted on U d nd ts grdent I I I I I ( u) = µ ν r s represented by c I µ ( ucxtu ( ; ) ) = Rµ ( µ µ ) + χ µ I c I ν ( ucxtu) R ( ) I r ν ( ; ) = ν ν ν + χ () ucxtu ( ; ) = Rrxt ( ) c( xtu ; ) u c( x t; u) = R c ( x) c( x) Ac ( x ; u) where c * µ (xt;u) s the soluton of the djont proble ()-() ν χ χ nd µ G χk ( u) = c ( x t; u) c( x t; u) dxdt µ Ω χ ν k k G. () ν Ω k = ( ; ) ( ; ) u c x t u c x t u dxdt For ny pont u * U * t s necessry the followng nequlty s fulflled ( * I u ) u u for ll u U d. U 5 Suffcent Condton of xstence of Sttonry Ponts In the prevous secton hs been shown tht the set of nu of the cost functonl (9) on U d s non-epty. But the pont of cost functonl nu cn be both nteror nd boundry pont of U d. It s cler tht n nteror pont of cost functonl nu s the sttonry pont. heore 4. Let stte functon c(xt;u) s governed by the frst ntl-boundry vlue proble (4) (6)-(7) the coeffcents of whch stsfy to restrctons (5). Let the set of dssble odel preters U d s convex closed bounded set nd the followng u = µν r c U d nequltes re vld for ll

10 68 S. Kvv r ν ν δ ν µ µ δ µ δ ( Q) r L c c c δ. W ( Ω) hen t s suffcent for exstence of unque sttonry pont on U d tht the weght functons R R R R µ R ν stsfed to the followng nequltes for ll uw Ud nd q ( ) R c ( x t; u) δr L ( Q) µ ( u) R µ χ δ µ R Ac ( x ; u) δ c W ( Ω) ν R ( u) ν χ δ ν R c ( xtu ; ) c ( xtw ; ) + R A c ( xu ; ) c ( xw ; ) + L ( Q) W ( Ω) ( µ µ χ χ ) χ ν χ ν µ ν + R u w + R u w q u w where c * (xt;u) s the soluton of the djont proble ()-() nd µ ν χ u χ u re defned by (). Relly we cn rewrte the sttonrty condtons () s the followng opertor equton u = F( u). It s obvously tht under ssuptons of heore 4 the opertor F s contrctng nd ppng the set U d nto tself. U References. Onsh Y. Serne J. Arnold. Cown C. hopson F.: Crtcl revew: rdonuclde trnsport sedent trnsport wter qulty thetcl odelng nd rdonuclde dsorpton/desorpton echns. NURG/CR-3. Pcfc Northwest Lbortory Rchlnd (98). Zheleznyk M. Dechenko R. Khursn S. Kuzenko Yu. klch P. Vtjuk N.: Mthetcl odelng of rdonuclde dsperson n the Prpyt-Dneper qutc syste fter the Chernobyl ccdent. he Scence of the otl nvronent Vol.. (99) Robnson A.R. Lerusux P.F.J. Slon III N.Q.: Dt Asslton. he se Vol.. John Wley & Sons Inc. (998) Penenko V.V.: Soe spects of thetcl odelng usng the odels together wth observtonl dt. Bull. Nov. Cop. Center Nu. Model. n Atosph. Ocen nd nvronent Studes Vol. 4. Coputer Center Novosbrsk (996) Ldyzhensky O.A. Solonnkov V.A. Urltsev N.N.: he lner nd quslner equtons of prbolc type. Nuk Moscow (967) (In Russn) 6. Ldyzhensky O.A.: A xed proble for hyperbolc equtons. Gostekhzdt Moscow (953) (In Russn)